ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI

Description of Riemann Type I Ellipsoids

Type I Riemann Ellipsoids

Example Equilibrium Model

This particular set of seven key parameters has been drawn from [EFE] Chapter 7, Table XIII (p. 170). The tabular layout presented here, also appears in a related discussion labeled, Challenges Pt. 2.

$a=a_1=1$

$b=a_2=1.25$

$c=a_3=0.4703$

${\mathrm{\Omega }}_2=0.3639$

${\mathrm{\Omega }}_3=0.6633$

${\zeta }_2=-2.2794$

${\zeta }_3=-1.9637$

As a consequence — see an accompanying discussion (alternatively, ChallengesPt6) for details — the values of other parameters are …

					Example Values
$\tan \theta$	$=$	$-\frac{{\zeta }_2}{{\zeta }_3}\left[\frac{a^2+b^2}{a^2+c^2}\right]\frac{c^2}{b^2}=-0.344793$			$\theta =$	$-19.0238^{\circ }$
$\mathrm{\Lambda }$	$\equiv$	$\left[\frac{a^2}{a^2+b^2}\right]{\zeta }_3\cos \theta -\left[\frac{a^2}{a^2+c^2}\right]{\zeta }_2\sin \theta$			$\mathrm{\Lambda }=$	$-1.332892$
$\frac{y_0}{z_0}$	$=$	$\left[\frac{a^2}{a^2+c^2}\right]\frac{{\zeta }_2}{\mathrm{\Lambda }}=-\frac{b^2\sin \theta }{(c^2{\cos }^2\theta +b^2{\sin }^2\theta )}$			$\frac{y_0}{z_0}=$	$+1.400377$
$\frac{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}{y_{\mathrm{m}\mathrm{a}\mathrm{x}}}$	$=$	$\left\{\mathrm{\Lambda }\right[\frac{a^2+b^2}{b^2}]\frac{\cos \theta }{{\zeta }_3}{\}}^{1/2}$			$\frac{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}{y_{\mathrm{m}\mathrm{a}\mathrm{x}}}=$	$+1.025854$
	$=$	$\frac{(c^2{\cos }^2\theta +b^2{\sin }^2\theta {)}^{1/2}}{bc}$
$\dot{\phi }$	$=$	$\left\{\mathrm{\Lambda }\right[\frac{b^2}{a^2+b^2}]\frac{{\zeta }_3}{\cos \theta }{\}}^{1/2}$			$\dot{\phi }=$	$+1.299300$

EFE Rotating Cartesian Frame

Concentric triaxial ellipsoids are defined by the expression,

$P$

$=$

$(\frac{x}{a}{)}^2+(\frac{y}{b}{)}^2+(\frac{z}{c}{)}^2,$

where $0\le P\le 1$ is a constant. As viewed from the rotating reference frame, the velocity flow-field everywhere inside $(0\le P<1)$, and on the surface $(P=1)$ of the Type I Riemann ellipsoid is given by the expression — see, for example, an accompanying discussion of the Riemann flow-field,

${\mathbf{𝐮}}_{\mathrm{E}\mathrm{F}\mathrm{E}}$

$=$

$\hat{\mathit{ı}}\{-[\frac{a^2}{a^2+b^2}]{\zeta }_{3}y+[\frac{a^2}{a^2+c^2}\left]{\zeta }_{2}z\right\}+\hat{\mathit{ȷ}}\{+[\frac{b^2}{a^2+b^2}\left]{\zeta }_{3}x\right\}+\hat{\mathbf{k}}\{-[\frac{c^2}{a^2+c^2}\left]{\zeta }_{2}x\right\}.$

In an accompanying discussion, we have shown that,

${\mathbf{𝐮}}_{\mathrm{E}\mathrm{F}\mathrm{E}}\cdot \mathrm{\nabla }P$

$=$

$0,$

which means that, at every location inside and on the surface of the configuration, the velocity vector is orthogonal to a vector that is normal to the ellipsoidal surface at that location.

Tilted Coordinate System

Figure 1:   Tilted Reference Frame
$\hat{\mathit{ı}}$ $\to$ ${\hat{\mathit{ı}}}^{\prime },$ $\hat{\mathit{ȷ}}$ $\to$ ${\hat{\mathit{ȷ}}}^{\prime }\cos \theta -{\hat{\mathit{k}}}^{\prime }\sin \theta ,$ $\hat{\mathit{k}}$ $\to$ ${\hat{\mathit{ȷ}}}^{\prime }\sin \theta +{\hat{\mathit{k}}}^{\prime }\cos \theta .$		$x$ $\to$ $x^{\prime },$ $y$ $\to$ $y^{\prime }\cos \theta -z^{\prime }\sin \theta ,$ $z-z_0$ $\to$ $y^{\prime }\sin \theta +z^{\prime }\cos \theta .$

As we have detailed in our accompanying discussion, as viewed from this "tipped" frame, the concentric ellipsoidal surfaces of a Type I Riemann ellipsoid are defined by the expression,

$P^{\prime }$

$=$

$[\frac{y^{\prime }\cos \theta -z^{\prime }\sin \theta }{b}{]}^2+[\frac{z_0+z^{\prime }\cos \theta +y^{\prime }\sin \theta }{c}{]}^2+(\frac{x^{\prime }}{a}{)}^2.$

and the velocity flow-field is given by the expression,

${{\mathit{u}}^{\prime }}_{\mathrm{E}\mathrm{F}\mathrm{E}}$	$=$	${\hat{\mathit{ı}}}^{\prime }\{-[\frac{a^2}{a^2+b^2}]{\zeta }_3(y^{\prime }\cos \theta -z^{\prime }\sin \theta )+[\frac{a^2}{a^2+c^2}]{\zeta }_2(z_0+y^{\prime }\sin \theta +z^{\prime }\cos \theta )\}$
		$+[{\hat{\mathit{ȷ}}}^{\prime }\cos \theta -{\hat{\mathbf{k}}}^{\prime }\sin \theta ]\left\{\right[\frac{b^2}{a^2+b^2}\left]{\zeta }_3{x}^{\prime }\right\}+[{\hat{\mathit{ȷ}}}^{\prime }\sin \theta +{\hat{\mathbf{k}}}^{\prime }\cos \theta ]\{-[\frac{c^2}{a^2+c^2}\left]{\zeta }_2{x}^{\prime }\right\}.$

We also have explicitly demonstrated that, for any arbitrarily chosen value of the tilt angle, $\theta$,

${{\mathbf{𝐮}}^{\prime }}_{\mathrm{E}\mathrm{F}\mathrm{E}}\cdot \mathrm{\nabla }{P}^{\prime }$

$=$

$0.$

Preferred Tilt

As we discuss elsewhere, if we specifically choose,

$\tan \theta$

$=$

$-\frac{\beta {\mathrm{\Omega }}_2}{\gamma {\mathrm{\Omega }}_3}=-\left[\frac{c^2(a^2+b^2)}{b^2(a^2+c^2)}\right]\frac{{\zeta }_2}{{\zeta }_3}.$

the component of the flow-field in the ${\hat{\mathbf{k}}}^{\prime }$ direction vanishes; that is, in this specific case, as viewed from the tilted reference frame, all of the fluid motion is confined to the x'-y' plane. Notice that this plane is not parallel to any of the three principal planes of the Type I Riemann ellipsoid. I have not seen this fluid-flow behavior previously described in the published literature. Maybe Norman Lebovitz will know.

The three panels of Figure 2, and the text description that follows, have been drawn from a separate discussion.

Figure 2a	Figure 2b
↲
Figure 2c
↲

As has been described in an accompanying discussion of Riemann Type 1 ellipsoids, we have used COLLADA to construct an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 2 show what this ellipsoid looks like when viewed from two different perspectives. (As a reminder — see the explanation accompanying Figure 2 of that accompanying discussion — the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25° to the equilibrium spin axis, which is shown in green.) Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the table that accompanies that discussion. From the perspective presented in Figure 2b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z₀ = -0.25, the smallest corresponds to our choice of z₀ = -0.60, and the one of intermediate size correspond to our choice of z₀ = -0.4310. When viewed from the perspective presented in Figure 2a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of θ = -19.02° to the untilted equatorial, x-y plane of the purple ellipsoid.

See Also

• Description of Riemann Type I Ellipsoids
• Riemann Type 1 Ellipsoids (old introduction)
• Construction Challenges (Pt. 1)
• Construction Challenges (Pt. 2)
• Construction Challenges (Pt. 3)
• Construction Challenges (Pt. 4)
• Construction Challenges (Pt. 5)
• Construction Challenges (Pt. 6)

• Related discussions of models viewed from a rotating reference frame:
  • PGE
  • NOTE to Eric Hirschmann & David Neilsen... I have moved the earlier contents of this page to a new Wiki location called Compressible Riemann Ellipsoids.